Free probability

Free probability is a branch of mathematics that studies how certain random-like objects, called "non-commutative random variables," behave when combined, especially when they are independent in a special sense called "freeness." Unlike traditional probability, which deals with regular numbers and their averages, free probability focuses on operators or matrices that do not commute (the order of multiplication matters). It provides tools to analyze large complex systems in fields like quantum physics and random matrix theory, helping us understand the collective behavior of many interacting components in a rigorous mathematical framework.